Tables for
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.2, pp. 740-741

Section Optical properties

H. Klappera and Th. Hahna Optical properties

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Optical studies provide good facilities with which to determine the symmetry of transparent crystals. The following optical properties may be used. Refraction

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The dependence of the refractive index on the vibration direction of a plane-polarized light wave travelling through the crystal can be obtained from the optical indicatrix. This surface is an ellipsoid, which can degenerate into a rotation ellipsoid or even into a sphere. Thus, the lowest symmetry of the property `refraction' is 2/m 2/m 2/m, the point group of the general ellipsoid. According to the three different forms of the indicatrix, three categories of crystal systems have to be distinguished (Table[link]).

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Categories of crystal systems distinguished according to the different forms of the indicatrix

Crystal systemShape of indicatrixOptical character
Cubic Sphere Isotropic (not doubly refracting)
[\!\left.\matrix{\hbox{Tetragonal}\hfill\cr \hbox{Trigonal}\hfill\cr \hbox{Hexagonal}\hfill\cr}\right\}\hfill] Rotation ellipsoid [\!\left.\matrix{\hbox{Uniaxial}\hfill\cr \noalign{\vskip 2pt}\cr \hfill\cr \hbox{Biaxial}\hfill\cr}\right\}\matrix{\hbox{Anisotropic}\hfill\cr\quad\hbox{(doubly}\hfill\cr\quad\hbox{refracting)}\hfill\cr}]
[\!\left.\matrix{\hbox{Orthorhombic}\hfill\cr\hbox{Monoclinic}\hfill\cr\hbox{Triclinic}\hfill\cr}\right\}] General ellipsoid

The orientation of the indicatrix is related to the symmetry directions of the crystal. In tetragonal, trigonal and hexagonal crystals, the rotation axis of the indicatrix (which is the unique optic axis) is parallel to the main symmetry axis. For ortho­rhombic crystals, the three principal axes of the indicatrix are oriented parallel to the three symmetry directions of the crystal. In the monoclinic system, one of the axes of the indicatrix coincides with the monoclinic symmetry direction, whereas in the triclinic case, the indicatrix can, in principle, have any orientation relative to a chosen reference system. Thus, in triclinic and, with restrictions, in monoclinic crystals, the orientation of the indicatrix can change with wavelength λ and temperature T (orientation dispersion). In any system, the size of the indicatrix and, in all but the cubic system, its shape can also vary with λ and T.

When studying the symmetry of a crystal by optical means, note that strain can lower the apparent symmetry owing to the high sensitivity of optical properties to strain. Optical activity

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The symmetry information obtained from optical activity is quite different from that given by optical refraction. Optical activity is in principle confined to the 21 noncentrosymmetric classes but it can occur in only 15 of them (Table[link]). In the 11 enantiomorphism classes, a single crystal is either right- or left-handed. In the four non-enantiomorphous classes [m,\ mm2,\ \overline{4}] and [\overline{4}2m], optical activity may also occur; here directions of both right- and left-handed rotations of the plane of polarization exist in the same crystal. In the other six noncentrosymmetric classes, [3m], [4mm, \overline{6}, 6mm, \overline{6}2m, \overline{4}3m], optical activity is not possible.

In the two cubic enantiomorphous classes 23 and 432, the optical activity is isotropic and can be observed along any direction.23 For the other optically active crystals, the rotation of the plane of polarization can, in practice, be detected only in directions parallel (or approximately parallel) to the optic axes. This is because of the dominating effect of double refraction. No optical activity, however, is present along an inversion axis or along a direction parallel or perpendicular to a mirror plane. Thus, no activity occurs along the optic axis in crystal classes [\overline{4}] and [\overline{4}2m]. In classes m and mm2, no activity can be present along the two optic axes if these axes lie in m. If they are not parallel to m, they show optical rotation(s) of opposite sense. Second-harmonic generation (SHG)

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Light waves passing through a noncentrosymmetric crystal induce new waves of twice the incident frequency. This second-harmonic generation is due to the nonlinear optical susceptibility. The second-harmonic coefficients form a third-rank tensor, which is subject to the same symmetry constraints as the piezoelectric tensor (see Section[link]). Thus, only 20 noncentrosymmetric crystals, except those of class 432, can show the second-harmonic effect; cf. Table[link].

Second-harmonic generation is a powerful method of testing crystalline materials for the absence of a symmetry centre. With an appropriate experimental device, very small amounts (less than 10 mg) of powder are sufficient to detect the second-harmonic signals, even for crystals with small deviations from centrosymmetry (Dougherty & Kurtz, 1976[link]).


Dougherty, J. P. & Kurtz, S. K. (1976). A second harmonic analyzer for the detection of non-centrosymmetry. J. Appl. Cryst. 9, 145–158.

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